app1_reli_final

app1_reli_final - Application Example 1 (Probability of...

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Application Example 1 (Probability of combinations of events; binomial and Poisson distributions) RELIABILITY OF SYSTEMS WITH VARIOUS ELEMENT CONFIGURATIONS Note: Sections 1, 3 and 4 of this application example require only knowledge of events and their probability. Section 2 involves the binomial and Poisson distributions. 1: SERIES AND PARALLEL SYSTEMS Many physical and non-physical systems (e.g. bridges, car engines, air-conditioning systems, biological and ecological systems, chains of command in civilian or military organizations, quality control systems in manufacturing plants, etc.) may be viewed as assemblies of many interacting elements. The elements are often arranged in mechanical or logical series or parallel configurations. Series systems Series systems function properly only when all their components function properly. Examples are chains made out of links, highways that may be closed to traffic due to accidents at different locations, the food chains of certain animal species, and layered company organizations in which information is passed from one hierarchical level to the next.
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The reliability of a series system is easily calculated from the reliability of its components. Let Y i be an indicator of whether component i fails or not; hence Y i = 1 if component i fails and Y i = 0 if component i functions properly. Also denote by P i = P[Y i = 1] the probability that component i fails. The probability of failure of a system with n components in series is then P[system failure] = 1 P[system survival] = 1 P[(Y 1 = 0) (Y 2 = 0) ... (Y n = 0)] (1) If the components fail or survive independently of one another, then this probability becomes n P[system failure] = 1 (1 P i ) (2) i = 1 In the even more special case when the component reliabilities are all the same, P i = P and Eq. 2 gives P[system failure] = 1 (1 P) n (3) Parallel systems In this case, the system fails only if all its components fail. For example, if an office has n copy machines, it is possible to copy a document if at least one machine is in good working conditions. Schematic illustration of a parallel system 2
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The probability of failure of a parallel system of this type is obtained as P[system failure] = P[(Y 1 = 1) (Y 2 = 1) ... (Y n = 1)] n = P i , if the components fail independently (4) i = 1 = P n , if in addition P i = P for all i Problem 1.1 Consider a series system. Plot its probability of failure in Eq. 3 as a function of the number of components n, for different values of P. Do the same for parallel systems, using the last expression in Eq. 4. Comment on the effect of n in the two cases. 2: m-out-of-n SYSTEMS Simple series and parallel representations are often inadequate to describe real systems. A first generalization, which includes series and parallel systems as extreme cases, is that
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app1_reli_final - Application Example 1 (Probability of...

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